2023 Fiscal Year Final Research Report
Toward the understanding of organizing center for the complex dynamics in dissipative systems
| Project/Area Number |
20K20341
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| Project/Area Number (Other) |
18H05322 (2018-2019)
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| Research Category |
Grant-in-Aid for Challenging Research (Pioneering)
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| Allocation Type | Multi-year Fund (2020)
Single-year Grants (2018-2019) |
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| Review Section |
Medium-sized Section 12:Analysis, applied mathematics, and related fields
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| Research Institution | Hokkaido University (2019-2023)
Tohoku University (2018) |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
藪 浩 東北大学, 材料科学高等研究所, 教授 (40396255)
渡辺 毅 公立諏訪東京理科大学, 工学部, 特任准教授 (40726676)
國府 寛司 京都大学, 理学研究科, 教授 (50202057)
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| Project Period (FY) |
2020-04-01 – 2024-03-31
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| Keywords | 大域分岐解析 / ナノ微粒子 / 反応拡散方程式 / 相分離 / 自己組織化 / Cahn-Hilliard 方程式 / 空間局在解 |
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| Outline of Final Research Achievements |
We investigated the diverse structural formations within nano-particles using the coupled Cahn-Hilliard equations. Through a systematic numerical exploration, we revealed the significance of hierarchical saddle solution networks embedded in the corresponding infinite-dimensional free energy landscape. Additionally, we found that the formation of counterintuitive polyhedral structures is deeply related to experimental initial settings such as pressure and initial concentration of polymers, which can be controlled by the characteristic time scales of the model equations. This time scale control represents a new dynamic approach complementing static "universal cells" viewpoint.Moreover, our perspective from the viewpoint of non-equilibrium organizing center such as high Morse index singularities has provided key contribution to understand the complex pattern dynamics arising in 2D collision problems and dynamics of patterns with oscillatory tails in heterogeneous media.
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| Free Research Field |
応用数学
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| Academic Significance and Societal Importance of the Research Achievements |
無限次元自由エネルギーの探索方法の新たな視座を提供した．自由エネルギーには膨大な極小解が存在し，それらはサドル型分水嶺により分たれている．その網羅的探索は困難であるが，高い不安定性を持つサドル解の成すネットワークとその下流探索により，興味ある解のクラスを網羅的に列挙できることが可能となった．また未知関数が複数である場合にはその時定数比の変化が分たれたbasin探索に極めて有効であることが示された．実際，時定数変化は実験設定と密接に関係しており，材料は同じでも実験状況に応じて様々な生成物が得られることと関連している．この方法は普遍的に適用可能であり，実験家にとっての羅針盤としての役割を果たす．
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